Integral of (-x+3/2)^2 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \frac{3}{2} - x$.

      Then let $du = - dx$ and substitute $- du$:

      $$\int \left(- u^{2}\right)\, du$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int u^{2}\, du = - \int u^{2}\, du$$

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int u^{2}\, du = \frac{u^{3}}{3}$$

         So, the result is: $- \frac{u^{3}}{3}$

      Now substitute $u$ back in:

      $$- \frac{\left(\frac{3}{2} - x\right)^{3}}{3}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(\frac{3}{2} - x\right)^{2} = x^{2} - 3 x + \frac{9}{4}$$

   2. Integrate term-by-term:

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- 3 x\right)\, dx = - 3 \int x\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x\, dx = \frac{x^{2}}{2}$$

         So, the result is: $- \frac{3 x^{2}}{2}$

      1. The integral of a constant is the constant times the variable of integration:

         $$\int \frac{9}{4}\, dx = \frac{9 x}{4}$$

      The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + \frac{9 x}{4}$

2. Now simplify:

   $$\frac{\left(2 x - 3\right)^{3}}{24}$$

3. Add the constant of integration:

   $$\frac{\left(2 x - 3\right)^{3}}{24}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(2 x - 3\right)^{3}}{24}+ \mathrm{constant}$$